Logic Seminar
Maxwell Levine
KGRC, Vienna, Austria
Patterns of Stationary Reflection
Abstract: Singular cardinals yield surprising results in set theory. After Cohen proved that CH is independent of ZFC, Easton proved that on regular cardinals, the continuum function $\kappa \mapsto 2^\kappa$ is constrained only by the facts that $\lambda \le \kappa \Rightarrow 2^\lambda \le 2^\kappa$ and that $\operatorname{cf}(2^\kappa)>\kappa$. In other words, the ZFC constraints on $\kappa \mapsto 2^\kappa$ are fully characterized relative to the class of regular cardinals. In an unexpected turn, Silver proved that GCH cannot fail for the first time at a singular cardinal of uncountable cofinality. More constraints on the arithmetic of singular cardinals were later discovered by Shelah using his PCF theory.
This opens up a more general scheme of questions: Given a property $P$ of a cardinal $\kappa$, what can ZFC prove about the behavior of $P(\kappa)$ across the class of all cardinals? Does $\{P(\kappa):\kappa<\lambda\}$ ever imply $P(\lambda)$?
For this talk, we will present an Easton-style result for stationary reflection. If $S$ is a stationary subset of a cardinal $\kappa$, the reflection principle $SR(S)$ asserts that every stationary subset of $S$ reflects. Assuming the consistency of a supercompact cardinal, we prove that given a fixed $n<\omega$, there are only a few trivial ZFC constraints on $SR(\kappa \cap \operatorname{cof}(\aleph_n))$ (current work in inner model theory suggests that the large cardinal assumption is close to optimal). The successors of singular cardinals present the greatest hurdle for this result, and require a nonstandard approach to PCF theory.
This is joint work with Sy-David Friedman.
Tuesday October 1, 2019 at 3:30 PM in 427 SEO